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发表于 2011-5-15 19:32:32
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来自 四川成都
我计算了一个两个参数的最小二乘的例子:
准确值为20,40;
有100组测量数据,其中c为系数矩阵,d为准确解,d1为扰动d的正负2%扰动值,所求参数为大于零作为约束条件。同时计算cond(c)=76.1092,也就是说系数矩阵的条件数比较小,扰动影响较小,三种方法都可以。
程序如下:
c=[0.0000E+00 0.0000E+00
1.6442E-07 3.3162E-07
6.4651E-07 1.3153E-06
1.4296E-06 2.9344E-06
2.4969E-06 5.1721E-06
3.8317E-06 8.0117E-06
5.4173E-06 1.1437E-05
7.2370E-06 1.5430E-05
9.2741E-06 1.9975E-05
1.1512E-05 2.5055E-05
1.3933E-05 3.0654E-05
1.6522E-05 3.6754E-05
1.9262E-05 4.3339E-05
2.2135E-05 5.0392E-05
2.5125E-05 5.7896E-05
2.8215E-05 6.5836E-05
3.1389E-05 7.4193E-05
3.4630E-05 8.2952E-05
3.7921E-05 9.2095E-05
4.1246E-05 1.0161E-04
4.4587E-05 1.1147E-04
4.7931E-05 1.2166E-04
5.1275E-05 1.3218E-04
5.4619E-05 1.4299E-04
5.7963E-05 1.5409E-04
6.1307E-05 1.6546E-04
6.4651E-05 1.7708E-04
6.7995E-05 1.8893E-04
7.1339E-05 2.0100E-04
7.4683E-05 2.1327E-04
7.8027E-05 2.2572E-04
8.1371E-05 2.3834E-04
8.4716E-05 2.5111E-04
8.8060E-05 2.6402E-04
9.1404E-05 2.7704E-04
9.4748E-05 2.9016E-04
9.8092E-05 3.0337E-04
1.0144E-04 3.1664E-04
1.0478E-04 3.2997E-04
1.0812E-04 3.4332E-04
1.1147E-04 3.5670E-04
1.1481E-04 3.7007E-04
1.1816E-04 3.8345E-04
1.2150E-04 3.9683E-04
1.2484E-04 4.1020E-04
1.2819E-04 4.2358E-04
1.3153E-04 4.3695E-04
1.3488E-04 4.5033E-04
1.3822E-04 4.6371E-04
1.4156E-04 4.7708E-04
1.4491E-04 4.9046E-04
1.4825E-04 5.0383E-04
1.5160E-04 5.1721E-04
1.5494E-04 5.3059E-04
1.5828E-04 5.4396E-04
1.6163E-04 5.5734E-04
1.6497E-04 5.7072E-04
1.6832E-04 5.8409E-04
1.7166E-04 5.9747E-04
1.7500E-04 6.1084E-04
1.7835E-04 6.2422E-04
1.8169E-04 6.3760E-04
1.8504E-04 6.5097E-04
1.8838E-04 6.6435E-04
1.9172E-04 6.7772E-04
1.9507E-04 6.9110E-04
1.9841E-04 7.0448E-04
2.0176E-04 7.1785E-04
2.0510E-04 7.3123E-04
2.0844E-04 7.4460E-04
2.1179E-04 7.5798E-04
2.1513E-04 7.7136E-04
2.1848E-04 7.8473E-04
2.2182E-04 7.9811E-04
2.2516E-04 8.1149E-04
2.2851E-04 8.2486E-04
2.3185E-04 8.3824E-04
2.3520E-04 8.5161E-04
2.3854E-04 8.6499E-04
2.4189E-04 8.7837E-04
2.4523E-04 8.9174E-04
2.4857E-04 9.0512E-04
2.5192E-04 9.1849E-04
2.5526E-04 9.3187E-04
2.5861E-04 9.4525E-04
2.6195E-04 9.5862E-04
2.6529E-04 9.7200E-04
2.6864E-04 9.8538E-04
2.7198E-04 9.9875E-04
2.7533E-04 1.0121E-03
2.7867E-04 1.0255E-03
2.8201E-04 1.0389E-03
2.8536E-04 1.0523E-03
2.8870E-04 1.0656E-03
2.9205E-04 1.0790E-03
2.9539E-04 1.0924E-03
2.9873E-04 1.1058E-03
3.0208E-04 1.1191E-03
3.0542E-04 1.1325E-03
3.0877E-04 1.1459E-03
3.1211E-04 1.1593E-03];
d=[0.0000E+00
1.6553E-05
6.5543E-05
1.4597E-04
2.5682E-04
3.9710E-04
5.6581E-04
7.6194E-04
9.8448E-04
1.2324E-03
1.5048E-03
1.8006E-03
2.1188E-03
2.4584E-03
2.8184E-03
3.1977E-03
3.5955E-03
4.0107E-03
4.4422E-03
4.8891E-03
5.3505E-03
5.8252E-03
6.3126E-03
6.8121E-03
7.3230E-03
7.8445E-03
8.3761E-03
8.9171E-03
9.4667E-03
1.0024E-02
1.0589E-02
1.1161E-02
1.1739E-02
1.2322E-02
1.2910E-02
1.3502E-02
1.4097E-02
1.4694E-02
1.5294E-02
1.5895E-02
1.6497E-02
1.7099E-02
1.7701E-02
1.8303E-02
1.8905E-02
1.9507E-02
2.0109E-02
2.0711E-02
2.1313E-02
2.1915E-02
2.2516E-02
2.3118E-02
2.3720E-02
2.4322E-02
2.4924E-02
2.5526E-02
2.6128E-02
2.6730E-02
2.7332E-02
2.7934E-02
2.8536E-02
2.9138E-02
2.9740E-02
3.0342E-02
3.0943E-02
3.1545E-02
3.2147E-02
3.2749E-02
3.3351E-02
3.3953E-02
3.4555E-02
3.5157E-02
3.5759E-02
3.6361E-02
3.6963E-02
3.7565E-02
3.8167E-02
3.8769E-02
3.9370E-02
3.9972E-02
4.0574E-02
4.1176E-02
4.1778E-02
4.2380E-02
4.2982E-02
4.3584E-02
4.4186E-02
4.4788E-02
4.5390E-02
4.5992E-02
4.6594E-02
4.7195E-02
4.7797E-02
4.8399E-02
4.9001E-02
4.9603E-02
5.0205E-02
5.0807E-02
5.1409E-02
5.2011E-02
5.2613E-02];
x0=cond(c)
x1=c\d
[x2,resnorm]=lsqnonneg(c,d)
[x3,resnorm]=lsqlin(c,d,[],[],[],[],[0,0],[],[],optimset('largescale','off'))
d1=[0.0000E+00
1.6666E-05
6.6277E-05
1.4888E-04
2.5274E-04
4.0257E-04
5.7015E-04
7.7125E-04
9.9420E-04
1.2119E-03
1.4779E-03
1.7881E-03
2.0814E-03
2.4787E-03
2.8283E-03
3.1648E-03
3.6609E-03
4.0902E-03
4.3830E-03
4.9623E-03
5.3321E-03
5.7703E-03
6.2404E-03
6.6801E-03
7.2150E-03
7.7428E-03
8.3812E-03
8.7934E-03
9.3299E-03
1.0217E-02
1.0743E-02
1.1204E-02
1.1903E-02
1.2479E-02
1.3021E-02
1.3501E-02
1.4138E-02
1.4715E-02
1.5439E-02
1.5698E-02
1.6691E-02
1.7092E-02
1.7906E-02
1.8186E-02
1.8644E-02
1.9661E-02
2.0309E-02
2.0965E-02
2.0952E-02
2.2202E-02
2.2786E-02
2.2878E-02
2.3453E-02
2.4682E-02
2.5225E-02
2.5085E-02
2.5660E-02
2.6320E-02
2.7083E-02
2.7461E-02
2.8134E-02
2.9522E-02
2.9265E-02
3.0861E-02
3.1410E-02
3.1882E-02
3.2749E-02
3.3077E-02
3.3327E-02
3.3300E-02
3.4838E-02
3.5089E-02
3.6281E-02
3.6344E-02
3.7092E-02
3.7175E-02
3.8053E-02
3.8558E-02
3.8833E-02
3.9977E-02
4.0400E-02
4.0724E-02
4.2032E-02
4.2463E-02
4.3024E-02
4.3557E-02
4.4106E-02
4.4534E-02
4.4680E-02
4.6663E-02
4.6061E-02
4.7625E-02
4.8469E-02
4.8792E-02
4.9196E-02
5.0261E-02
5.0882E-02
4.9843E-02
5.0933E-02
5.1782E-02
5.3325E-02];
x4=c\d1
[x5,resnorm]=lsqnonneg(c,d1)
[x6,resnorm]=lsqlin(c,d1,[],[],[],[],[0,0],[],[],optimset('largescale','off'))
x0 =
76.1092
x1 =
20.0014
39.9996
x2 =
20.0014
39.9996
resnorm =
2.2548e-011
Optimization terminated.
x3 =
20.0014
39.9996
resnorm =
2.2548e-011
x4 =
19.3635
40.1961
x5 =
19.3635
40.1961
resnorm =
1.0015e-005
Optimization terminated.
x6 =
19.3635
40.1961
resnorm =
1.0015e-005
得出结论上述三种方法都该问题都有效。 |
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